The reaction is very slow to process.They're not really similar in acidity. The phenol proton has a pKa of about 10 and acetic acid has a pKa of about 5. Regardless, acidity is only one aspect of reactivity and cannot be used as a general guide to what compounds will and will not react.
If they were to react, they would form an ester. However, the reaction is known to be slow, and the resulting ester would be prone to hydrolysis.